The scaleless spin network state for a three-valent graph can be expressed into a Gaussian integral [1]}, [2]}. Let us introduce an auxiliary spinor \(|\xi _{nl}\rangle =\left( \begin{array}{c}\xi _{nl}^0\\ \xi _{nl}^1\end{array} \right)  \in {\mathbb {C}}^2\)  attached to each half link \(l\)  incident to node \(n\)  with the same spinor measure \(\mathrm {d}\mu (\xi _{nl})={\pi ^2}\mathrm {d}\xi _{nl}^0\mathrm {d}\bar{\xi }_{nl}^0\mathrm {d}\xi _{nl}^1\mathrm {d}\bar{\xi }_{nl}^1 e^{-\langle \xi _{nl}|\xi _{nl}\rangle }\) . The dual auxiliary spinor is defined in the same way as \(z\) 's, \(\textit {i.e.}\)  \(|\xi _{nl}]=\left( \begin{array}{c}-\bar{\xi }_{nl}^1\\ \bar{\xi }_{nl}^0\end{array} \right) \) .
To unify the notation, we denote the new spinors at the source \(s(l)\)  and target \(t(l)\)  of a link \(l\)  as \(\xi _l\)  and \(\tilde{\xi }_l\) .
Consider a general 3-cell \(\triangle \)  whose boundary \(\partial \triangle \)  is made up by jointed triangles whose dual graph \((\partial \triangle )^*_1\)  is a (closed) three-valent graph. The correspondent scaleless spin network state is [2]}
\({\mathcal {S}}^{\text{sl}}_{(\partial \triangle )^*_1}(\lbrace z_l,\tilde{z}_l\rbrace )=\int \left( \prod _{l} \mathrm {d}\mu (\xi _{l})\mathrm {d}\mu (\tilde{\xi }_{l}) \right)e^{\sum _{l} \langle \xi _{l}|\tilde{\xi }_{l}] + \sum _{\alpha } [z_{s(\alpha )}|z_{t(\alpha )}\rangle [\xi _{s(\alpha )}|\xi _{t(\alpha )}\rangle }\,,\) 
